The complex Lorentzian Leech lattice and the bimonster. II. (Q2790619)

The paper under review is to study the reflection group of the complex Lorentzian Leech lattice. Let \(\omega=e^{2\pi i/3}\) and \(\mathcal E=\mathbb Z[\omega]\). Define the \(\mathcal E\)-lattice \(L\) to be the direct sum of the complex Leech lattice and a hyperbolic cell. Let \(Y=\mathbb P_+(L^\mathbb C)\) be the set of complex lines of positive norm in the complex hyperbolic space \(L^{\mathbb C}=L\otimes_{\mathcal E}\mathbb C\). The projective automorphism group \(\Gamma=\mathbb P\Aut(L)\) acts faithfully on \(Y\). Let \(\mathcal M\) be the union of the fixed points of the reflections in \(\Gamma\). Let \(Y^\circ=Y\setminus\mathcal M\). Then \textit{D. Allcock} conjectured [in CRM Proceedings and Lecture Notes 47, 17-24 (2009; Zbl 1193.20015)] that the orbifold fundamental group of \(Y^\circ/\Gamma\) maps onto the bimonster (i.e., the wreath product of the monster simple group with \(\mathbb Z/2\mathbb Z\)).NEWLINENEWLINE The main result of the paper is that there is a homomorphism from the Artin group of the incident graph \(D\) of the projective plane over \(\mathbb F_3\) to the orbifold fundamental group of \(Y^\circ/\Gamma\), obtained by sending the Artin generators to the generators of monodromy around the mirrors of the generating reflections in \(\Gamma\). This takes a step towards the proof of Allcock's conjecture. The finite group \(\mathrm{PGL}(3,\mathbb F_3)\subseteq\Aut(D)\) acts on \(Y\) and fixes a complex hyperbolic line pointwise. The paper shows that the restriction of \(\Gamma\)-invariant meromorphic automorphic forms on \(Y\) to the complex hyperbolic line fixed by \(\mathrm{PGL}(3,\mathbb F_3)\) gives meromorphic modular forms of level 13.NEWLINENEWLINE For part I see [\textit{T. Basak}, J. Algebra 309, No. 1, 32-56 (2007; Zbl 1125.11040)].